DOI 10.1393/ncc/i2005-10109-2
Millisecond proto-magnetars and gamma ray bursts
(∗)
T. A. Thompson(∗∗)
Department of Astronomy & Theoretical Astrophysics Center
601 Campbell Hall, The University of California, Berkeley, CA 94720, USA (ricevuto il 23 Maggio 2005; pubblicato online il 2 Novembre 2005)
Summary. — In the seconds after core collapse and explosion, a thermal driven-neutrino wind emerges from the cooling, deleptonizing newly-born neutron star. If the neutron star has a large-scale magnetar-strength surface magnetic field and millisecond rotation period, then the wind is driven primarily by magneto-centrifugal slinging, and only secondarily by neutrino interactions. The strong magnetic field forces the wind to corotate with the stellar surface and the neutron star’s rotational energy is efficiently extracted. As the neutron star cools, and the wind becomes increasingly magnetically-dominated, the outflow becomes relativistic. Here I review the millisecond magnetar model for long-duration gamma ray bursts and explore some of the basic physics of neutrino-magnetocentrifugal winds. I further speculate on some issues of collimation and geometry in the millisecond magnetar model. PACS 97.60.Bw – Supernovae.
PACS 97.60.Jd – Neutron stars.
PACS 98.70.Rz – γ-ray sources; γ-ray bursts. PACS 97.10.Me – Mass loss and stellar winds. PACS 01.30.Cc – Conference proceedings.
1. – Introduction
A successful core-collapse supernova (SN) leaves behind a hot deleptonizing proto-neutron star (PNS) that cools and contracts on its Kelvin-Helmholtz cooling timescale (τKH ∼ 10 s),radiating its gravitational binding energy (∼ 1053ergs) in neutrinos [1]. A small fraction of these neutrinos deposit their energy in the tenuous atmosphere of the PNS through the interactionsνen → pe−, ¯νep → ne+, ν¯ν → e+e−,andν(e−e+)→ ν(e−e+). Inverse processes provide cooling. Net neutrino heating drives a thermal wind
that emerges into the post-supernova shock environment [2]. For typical non-rotating non-magnetic (NRNM) neutron stars,the total kinetic energy of the wind overτKHis of order ∼ 1048ergs and,hence,the addition to the asymptotic SN energetics is small on the scale of the canonical SN energy,ESN∼ 1051ergs [3-5].
Magnetars —a class of neutron stars with surface magnetic field strengths ofB0 ∼ 1015G— are thought to be born with millisecond rotation periods,their intense fields having been generated by an efficient dynamo [6-8]. Millisecond rotation periods im-ply a reservoir of rotational energy that is large on the scale of ESN: ERot ∼ 2 ×
(∗) Paper presented at the “4th Workshop on Gamma-Ray Burst in the Afterglow Era”, Rome, October 18-22, 2004.
(∗∗) Hubble fellow.
c
1052ergs M1.4R2ν10P1−2,where M1.4 = M/1.4 M, Rν10 = Rν/10 km,and P1 is the spin period in units of 1 millisecond (ms). Stellar progenitors that produce millisecond magnetars (MSMs) must have iron cores rotating with periods of ∼< 10 s at the moment of collapse [9]. The character of collapse,bounce,and explosion can be modified by rotation. For example,rotational support leads to characteristically lower neutrino luminosity (Lν) and longerτKH [10,11]. In addition, a fraction of the gravitational binding of rotating collapse may be stored in shear energy and tapped by viscous processes; for parameters appropriate to MSM birth this effect can yield explosions in models that would other-wise fail [11]. Although rotation may be important during collapse and explosion,and although small-scale magnetic fields may be important in providing a viscosity capable of tapping the shear energy generated during collapse,even the magnetic energy density as-sociated with a dipole field strength of 1015G is small with respect to the thermal pressure exterior to any PNS in the first∼ 1−2 seconds after collapse. Therefore,similar to NRNM PNSs,we expect that the wind that accompanies MSM cooling is driven by neutrino heat-ing at early times. However,as Lν decreases and the thermal pressure exterior to the MSM decreases,the region exterior to the MSM must become magnetically-dominated. The strong magnetic field forces the matter composing the outflow into near corotation with the stellar surface out to∼ RA,the Alfv´en point,where the magnetic energy density equals the kinetic energy density of the outflow. ForP ∼ 1 ms, if RA ∼> 15 km,then the wind is driven primarily by magneto-centrifugal slinging; neutrino heating becomes relatively unimportant in determining the asymptotic wind velocity. Rotational energy is transferred directly from the MSM to the wind and this provides an efficient mechanism for spindown [12,13]. The spin period of the MSMe-folds on the spindown timescale τJ ≈ (2/5)(M/ ˙M)(Rν/RA)2,where ˙M is the mass loss rate. Because ERot ESN,just onee-folding of Ω is sufficient to modify the dynamics of the SN remnant significantly. If τJ is small with respect to the time for the SN shock to traverse the progenitor (∼ tens
of seconds for type-Ib,-Ic progenitors) we also expect this extra energy source to modify the SN nucleosynthesis [14]. As the MSM cools and the outflow becomes increasingly magnetically-dominated,RA increases. It cannot do so indefinitely. RA approaches the radius of the light cylinderRL=c/Ω 48P1 km asymptotically. As it does so,the flow becomes increasingly relativistic. This is the transition between non-relativistic mass-loaded outflow and relativistic Poynting-flux dominated neutron star wind. All neutron stars,regardless of their initial spin period and magnetic field strength,go through this transition. MSMs are interesting because this transition occurs at high wind kinetic luminosity. Because ERot is large and the spindown timescale is short,and because the velocity of the wind must eventually become relativistic,these objects are a natural candidate for the central engines of long-duration gamma ray bursts (GRBs) [15,16,14].
2. – Proto-magnetar spindown
In this section I summarize the results of ref. [14]. See that work for more details. Angular momentum conservation implies that ˙J = d/dt(IΩ) = − ˙ML,where L is the specific angular momentum carried by the wind and I is the moment of inertia. In the classic model for solar spindown constructed by ref. [17],the wind problem is treated in one spatial dimension and in the equatorial plane. Consideration of the az-imuthal momentum equation together with Faraday’s law givesL = R2AΩ. To estimate the angular momentum loss rate and,thereby,the wind luminosity and the spindown timescale,we must first estimateRA. The location of the Alfv´en point depends on the radial dependence of the poloidal magnetic field. Because models with purely monopole
fields generally over-estimate spindown and models with pure dipole fields under-estimate spindown,we parameterizeBr=B0(Rν/r)η,where 2 ∼< η ∼< 3. Taking B2/8π ∼ ρvr2/2 atRA,ρ = ˙M/4πr2vr,and assuming thatvA=vr(RA)∼ vφ(RA)∼ RAΩ,we find that (1) RA=B02/(2η−1)R2η/(2η−1)ν ( ˙MΩ)−1/(2η−1),
wherevAis the radial Alfv´en speed,vris the radial velocity,vφ is the azimuthal velocity, ρ is the mass density,and Br is the radial magnetic field. Equation (1) assumes that RAΩ vν,where vν is the asymptotic wind velocity in a NRNM outflow (vν ∼< 3 × 109cm s−1). The absolute value of the rotational energy loss rate can be written as (2) E˙NR=B04/(2η−1)R4η/(2η−1)ν M˙(2η−3)/(2η−1)Ω(4η−4)/(2η−1).
For parameters appropriate to MSMs ˙ ENRη=2 1.5 × 1051B4/3 015R 8/3 ν10M˙−31/3P1−4/3ergs s−1 (3) ˙ ENRη=3 4.5 × 1050B4/5 015R 12/5 ν10 M˙−33/5P1−8/5ergs s−1. (4)
The subscript ‘NR’ is added to emphasize that when the flow is non-relativistic, ˙E depends explicitly on ˙M.
The non-relativistic scalings for the energy loss rate can be compared with those in the relativistic regime. AsRA becomes close toRL as ˙M decreases during the cooling epoch,vA approachesc and the flow becomes relativistic. At RL,the ratio of magnetic to kinetic energy density is
(5) Γ = B2 4πρc2 RL = B20Rν2ηΩ2η−2c1−2ηM˙−1
and the energy loss rate is
(6) E˙R=−Γ ˙Mc2=−B02R2ην Ω2η−2c3−2η.
Forη = 3 the classical “vacuum dipole” limit is obtained; ˙ER(η = 3) = B02R6νΩ4c−3(1). For the monopole case with η = 2, ˙ER is larger than the dipole limit by a factor of c2/(ΩR
ν)2— a factor of ∼ 23 for a 10 km MSM with a 1 ms spin period.
The non-relativistic spindown rate is larger than the relativistic spindown rate as a result of mass-loading. To see this explicitly,note that
˙ ENRη=2/ ˙Eη=2R 1 B0−2/315 R−4/3 ν10 M˙−31/3P12/3, (7) ˙ ENRη=3/ ˙Eη=3R 8 B0−6/515 R−18/5 ν10 M˙−33/5P112/5. (8)
For η = 2 we see that the ratio is approximately unity,reflecting the fact that for the parameters chosen RA ∼ RL. For slower spin periods,the ratio increases,but
(1) In the true “vacuum dipole” limit ˙ERhas a term sinα/6, where α is the angle between the spin axis and the magnetic dipole axis.
not dramatically. In contrast,the ratio of ˙ENR to ˙ER for the dipole case (η = 3) is large for MSMs and it has a strong dependence on P . Thus,for a magnetar born with a 10 ms spin period a naive application of the “vacuum dipole” formula underes-timates the magnitude of the rotational energy loss rate by a factor of∼ 2000. More detailed calculations reveal that when P ∼ 10 ms,for modest neutrino luminosities,
˙
M is probably closer to ∼ 10−5 M
s−1 so that the ratio of ˙ENR to ˙ER is closer to
∼ 120 than 2000. Even so, ˙ENR/ ˙ER is very large and an application of the relativistic
formula in an epoch when ˙M is large is incorrect. Of course,if the field structure is not purely dipolar (that is, η < 3),then the discrepancy between the “vacuum dipole” spindown approximation and the “true” spindown rate becomes even larger. In this case we would compare ˙ENRη=2/ ˙ERη=3 25B0−2/315 R−10/3ν10 M˙−31/3P18/3,a factor of 25 for a MSM.
For a magnetar with a 10 ms spin period and lower mass loss rate the ratio becomes ˙
ENRη=2/ ˙Eη=3R 2500B0−2/315 R−10/3ν10 M˙−51/3P108/3. These arguments serve to underscore the
fact that in the very early stages of proto-magnetar cooling,the multi-dimensional struc-ture of the wind/magnetic field interaction must be solved consistently (to determine the effectiveη) and that inferences about the “initial” spin period of magnetars should not be based on an application of the vacuum dipole approximation when ˙M is large.
For fixedB0, Rν,and Ω the transition from non-relativistic (eq. (4)) to relativistic (eq. (6)) outflow occurs when RA∼ RL and this point in time corresponds to a critical mass loss rate ˙Mcrit=B20Rν2ηΩ2η−2c1−2η,which scales withP−2andP−4 forη = 2 and η = 3,respectively. Because ˙M scales with the neutrino luminosity and because —to first approximation— the luminosity is a monotonically decreasing function of time,this scaling of ˙Mcrit with P implies that the wind is non-relativistic for a larger fraction of the cooling time for longer rotation periods: forη = 3 and P = 10 ms ˙Mcrit 3 × 10−9 Ms−1. Such a low ˙M may correspond to a time several tens of seconds after collapse.
3. – Are millisecond proto-magnetars GRB central engines?
The spindown timescale Ω/ ˙Ω in the non-relativistic limit is τJη=2NR 30 s M1.4M˙−3−1/3R−2/3ν10 B0−4/315 P −2/3 1 , (9) τJη=3NR 96 s M1.4M˙−3−3/5R−2/5ν10 B0−4/515 P −2/5 1 . (10)
For a MSM with 2× 1052ergs of rotational energy we need only wait a fraction ofτJto extract an amount of energy comparable to the supernova energy,∼ 1051ergs. Because τJ (or∼ τJ/5) is comparable to τKHwe infer that significant spindown may occur during
the cooling epoch. As implied by the discussion of ˙ENR above,for larger initial spin periods the spindown timescales decrease. In the relativistic limit (RA∼ RL)
τη=2 JR 34 s M1.4R−2ν10B −2 015, (11) τJη=3R 760 s M1.4R−4ν10B0−215P 2 1. (12)
AlthoughτJη=2R does not depend on Ω explicitly,it may have an implicit dependence on Ω if the magnetic field is generated by a dynamo [6,7].
With these timescales in hand we can (in a rudimentary way) attempt to assess the MSM GRB mechanism. A more detailed assessment must await a consistent multi-d MHD solution. From eqs. (3),(4),(9),and (10) we see that in the non-relativistic limit,
regardless of η,the amount of energy extractable on ∼ 10 − 100 second timescales is in the range appropriate to GRBs. For η < 3 these conclusions are stronger. Because ∼
> 1051ergs can be extracted on a timescale shorter than or comparable to the timescale
for the SN shock to traverse the progenitor,we expect that the wind may significantly affect the dynamics of the remnant and the 56Ni yield. In this way it may be possible to generate hyper-energetic or 1998bw-like SNe during MSM birth [14,18]. The inferred energetics and56Ni yield of SN2003dh and SN1998bw put strong constraints on any GRB mechanism. In the collapsar model a disk wind is thought to generate the56Ni required to power the SN lightcurve [19,20]. In the millisecond magnetar model, the energetic wind shocks the material already processed by the supernova shock,perhaps generating the large inferred56Ni yields; we are currently investigating the timing of this scenario.
Although it seems possible that MSM winds may generate energetic winds at early times,the flow during this mass-loaded wind phase —at least,on average— is not rel-ativistic. It is possible that a strong latitudinal dependence to the mass loss rate may yield relativistic asymptotic velocities for matter emerging from mid-latitudes even when our estimates would indicateRA< RL,but such a speculation must be tested against re-alistic multi-d models. It is also possible that large temporal variations in ˙M could cause the wind to alternate rapidly between non-relativistic and relativistic. Strong variations in the mass loading could be caused by shearing of large-scale closed magnetic loops on the surface of the fully convective MSM core [16].
As the flow becomes increasingly relativistic (on average),we see from eq. (12) that if the relativistic dipole limit strictly obtains,then the spindown timescale is long for B ∼ 1015G and P ∼ 1 ms. Although relativistic spindown with η = 3 will affect the
asymptotic remnant dynamics by injecting energy over a long timescale,it will probably not generate a GRB with duration ∼ 30 seconds and energy ∼ 1051ergs. Based on the scalings derived here,higher magnetic field strength,shorter spin period,or η < 3 is probably required in order for MSM spindown to power GRBs. If one of these possibilities obtains,then from the estimates in ref. [14] we find that essentially all of the magnetic energy at RL must be transferred to the wind in order for the flow to obtain high asymptotic Lorentz factor with large enough ˙E. This is presumably accomplished by magnetic dissipation [21,22].
4. – Emergence, geometry, and collimation
If a relativistic outflow with the requisite energy to power a GRB can be generated by a MSM,it must emerge from the massive stellar progenitor. The highly energetic non-relativistic wind,which precedes the relativistic outflow,will likely be collimated by hoop stress and will therefore shape the cavity into which the relativistic wind emerges. Because the non-relativistic wind carries little mass in comparison with the overlying star, the relativistic outflow is not additionally hindered in its escape from the progenitor by the preceding slow wind. Hence,if the relativistic outflow can be collimated,then the dynamics of its emergence from the progenitor should be qualitatively similar to models of collapsar jets escaping Type-Ibc progenitors [23].
One important possible objection to the MSM mechanism for GRBs is that it is difficult to collimate relativistic Poynting-flux dominated outflows [24]. Observational evidence for collimation in GRBs is abundant and so,at face-value,this would seem to be a problem. There are at least three responses to this objection. The first possible response is that the interaction between the emerging and energetic (non-relativistic and then relativistic) wind with the overlying post-supernova-explosion ejecta may act to
collimate the outflow. Future multi-d simulations should address this issue in detail. The second potential response is that,in fact,relativistic Poynting-flux dominated winds can be efficiently collimated,as in the work of ref. [25]. A final possible response is this: given the basic geometry of the magnetocentrifugal wind,it seems natural to suppose that the asymptotic radial velocity is largest in the equatorial region. This follows both from the fact that centrifugal acceleration is largest at the equator and that the equatorial current sheet may facilitate significant dissipation of the magnetic energy. Pulsars (e.g.,the Crab) provide evidence for high Lorentz factor and energetically-dominant equatorial winds [26,27]. In analogy, is it possible that the geometry of GRBs is “sheet”-like (equatorial) rather than jet-like? The solid angle subtended by a sheet with opening angleθ is ∼ θ,whereas for a jet it is ∼ θ2. This fact has implications for the afterglow —a sheet-break rather than a jet-break [28,29]— and for the GRB census. Inferences from detailed predictions of the flux evolution may be used to rule out or confirm the possibility of a sheet-like geometry in some bursts. I am constructing such models now.
∗ ∗ ∗
I thank P. Chang and E. Quataert for collaboration and J. Arons, B. Metzger, N. Bucciantini,and A. Spitkovsky for helpful conversations. Support for this work is provided by NASA through Hubble Fellowship grant #HST-HF-01157.01-A awarded by the Space Telescope Science Institute,which is operated by the Association of Universities for Research in Astronomy,Inc.,for NASA,under contract NAS 5-26555.
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